Longitudinal mode tuning of stringed instruments



United States Patent inventor Harold A. Conklin, Jr.

Cincinnati, Ohio Appl. No. 689,815

Filed Dec. 12, 1967 Patented Aug. 11, 1970 Assignee D. 1']. Baldwin Company Cincinnati, Ohio :1 Corp. of Ohio LONGITUDINAL MODE TUNING OF STRINGED INSTRUMENTS Primary Examiner- Richard B. Wilkinson Assistant Examiner John F. Gonzales Attorney-Melville, Strasser, Foster and Hoffman ABSTRACT: Stringed musical instruments in which the strings are tuned both for flexural modes of string vibration and for longitudinal modes of string vibration, the instrument and its strings being so designed that the frequencies of the fundamental longitudinal modes will bear a specific relationship to the frequencies of the fundamental flexural modes of string vibration.

LONGITUDINAL MODE TUNING OF STRINGED INSTRUMENTS BACKGROUND OF THE INVENTION This invention has to do with the design of the scales of stringed musical instruments with particular reference to the selection of the speaking lengths of the vibrating strings. While the invention will be described in reference to pianos, it should be understood from the outset that the benefits of the invention are not necessarily restricted to pianos but are applicable to other types of stringed instruments.

In conventional methods of scale design the speaking length of each string is determined in accordance with the following factors:

a. the fundamental flexural frequency at which the string must vibrate (in a piano such'frequ'ency would be determined by the pitch of the musicalnote corresponding to the instrument key with which the string is associated);

b. the desired string tension or stretching force, considering the cross-sectional area of the string and the physical constants and properties of the materials used to make the string;

0. the space available within the confines of the instrument for stretching the string, considering the layout of the scale, including the other strings and acoustical elements; and

d. the preferences of the individual designer concerning the manner in which the length of the strings should change from one note to the next.

in piano scales, the formula is the one normally employed by those skilled in the art to For any given note (frequency) of the scale, the length of the longest possible string is limited ultimately by the permissible tensile stress on the string material. A need to achieve the largest usable output of sound at the treble end of the scale (where the sound often tends to be undesirably weak) usually leads the designer to choose the longest practical string length for the note of highest frequency. For a standard piano having 88 notes, tuned to the standard pitch (where A =440 cycles per second) the nominal frequency of the highest note is 4186 cycles per second and, at the present state of the art, the practical string length (considering suitable safety factors) is limited by the tensile strength of the string material to only a little more than 2 inches even if the strongest steel piano wire is used. This limit is relatively independent of the crosssectional area of the string.

It is generally agreed that the quality of tone of the bass section of a piano improves as a string length is increased. However, the length of the bass strings is limited by the dimensions of the piano case. In order to provide good tone quality and to avoid sudden changes in tone quality from one note to the next note, it is a standard practice in piano design to make the length of the strings increase gradually from the treble to the bass end of the scale.

As those skilled in the art are aware, the designer of a piano scale us'ually attempts to maintain the tension or pull ofthe strings reasonably constant throughout the scale. If both the tension and the mass per unit length of the strings were to be kept constant throughout the scale, such a practice would in general result in an impractical design in that the bass strings would be of extreme length, being several times longer than those of the largest conventional pianos which provide space for strings'no longer't'han about 7 feetrconsequently, in order though they die away more quickly than the low-order flexural to maintain string tension reasonably constant throughout the scale, it is a customary practice for the designer to gradually increase not only the length of the strings but also the mass per unit length of the strings in proceeding from the treble to the bass end of the scale. The increase in mass per unit length is accomplished by progressive increases in the crosssectional area of the string wire. However, to increase the diameter of the strings beyond certain values, which depend upon the lengths of the strings and the materials from which they are formed, produces excessive string stiffness which has an undesirable effect on tone quality. Consequently, at some suitable point, the designer conventionally achieves any further necessary increase in mass per unit length of the strings by loading or wrapping them in helical fashion with a material, such as copper wire, instead of by using a plain string of larger diameter.

All of the foregoing considerations are based solely on the flexural mode of vibration of the string which is deemed to be the controlling mode of vibration in all conventional stringed instruments. The flexural mode of vibration is characterized by displacement of the string, under conditions of vibration, in a direction transverse to the direction of the string itself when it is at rest.

The existence of a longitudinal mode of string vibration, in which the direction of motion of the string elements during vibration is parallel to the direction of the string itself, has long been known but insofar as is known has not been considered significant in the design of stringed musical instruments and consequently has played no real part in their design.

SUMMARY OF THE INVENTION The present invention is based on the discovery that the iongitudinal mode of vibration of the strings plays a significant part in determining the characteristic sound of the instrumentv when-it is played in the normal way, and that if the scale of the instrument is designed so as to tune the frequencies of the lon-. gitudinal modes of vibration of the strings to certain optimum values for each string, improved tone quality and greater uniformity of tone quality from note to note over the scale of the instrument can be obtained. It has been further discovered that the tuning of the frequency of the longitudinal mode influences the clarity and definition of pitch, particularly in pianos having shorter strings, and that strings designed according to the principles set forth hereinafter produce a more pleasing sound and have a better defined pitch than those of conventional design, whether employed in small or large instruments. 1

It has been found that over a wide range of the keyboard,

what the ear hears when a piano note is sounded is in reality during which the sound resulting from the longitudinal mode of string vibration can be heard usually is considerably less than the time during which sound due to one or more flexural modes is audible. In many instances the chief contribution of the longitudinal mode consists of a transient tone most obvious at the time a note is first struck which dies away rather quickly thereafter. This tone nevertheless has been found to contribute importantly toward the over-all quality, color and pitch sensation of the entire tone, just as the higher order fiexural mode partials contribute importantly to the tone even mode partials. As persons who have a knowledge of music will be aware, the over-all quality, color, and texture ofa complete musical tone depends upon the cumulative effect of the frequencies, amplitudes, and temporal properties of allconstituents-of the tone. Certain combinations of ffequencies will create a' pleasing or harmonious sound, while others: will be musieallydiscordant'and disturbing'to the ear.

An object of this invention is the construction of a piano when successive adjacent notes are played on the keyboard the sound of the successive notes will have a similar characteristic sound so as to produce an improved degree of evenness in the scale of the instrument, evenness and uniformity of scale being one criterion for an excellent instrument.

It has been found that the tuning of the longitudinal mode has a significant effect upon the tone color of the instrument. and consequently a further object of the invention is to provide for more precise control of tone color through tuning of the longitudinal mode of string vibration, and to make possible the design of instruments especially suited to certain applications and to particular types or moods of music and to make the sound of the instruments more effective in evoking a variety ofemotional responses in the listener.

In accordance with the invention, the fundamental frequency of the longitudinal mode of each selected string will be tuned to a frequency closely corresponding to that ofa note of the equally tempered scale, with the longitudinal frequency of each string so tuned being separated from the fundamental flexural frequency of such string by a constant number of semitone intervals. More particularly, in the case of a piano having conventional steel strings, it has been found that tuning of the longitudinal mode to the frequency of a note of the equally tempered scale in the range 4000 5200 cents sharp compared to the fundamental flexural string frequency will produce highly preferred tone qualities. It is to be understood, however, that while the range given is a practical working range for pianos of conventional design and standard piano wire, the practical range may vary depending upon the circumstances. For example, the range will vary depending upo the type of stringed instrument being tuned.

It has been found, however, that irrespective of the range, the preferred tuning of the longitudinal modes will be at integral mutiples of approximately 100 cents sharp with respect to the fundamental flexural frequency of the string which is being tuned.

Even the foregoing considerations constitute an oversimplification due to the fact that the exact flexural frequencies of the strings of a properly tuned piano normally differ slightly from those of mathematically perfect equal temperament because of the inherent inharmonicity of the strings which influences the final flextiral string frequencies of the instrument during the tuning process. However, the necessary adjustments to compensate for inharmonicity can be readily calculated and will be explained in detail hereinafter.

DESCRIPTION OF THE PREFERRED EMBODIMENTS ment, usually one ofa pair of string-deflecting pins located on the soundboard bridge. When a piano hammer strikes. the

string it not only-induces vibration at the .flex'ural resonant. frequencies of the string, but also at the longitudinal frequen-. 'cies. The resulting vibration is transmitted to the soundboard of the instrument and some of this energy is radiated therefrom as sound energy.

BASIC CONSIDERATIONS.

The fundamental (lowest possible) resonant frequency of longitudinal vibration of a uniform homogeneous free rod depends upon the length of the rod and upon the elastic modulus and density (mass per unit volume) of the material of the rod. The textbook formula for the longitudinal frequency if.) of

such a rod is v b E 1/2 2L (P) HZ *eycles per second wherein E is the elastic or Youngs modulus of the material, P is the density (mass per unit volume) of the material, L is the length of the rod, and b is a constant whose value depends upon the system of units employed. Such a rod has other resonant frequencies. their values being, 2, 3, 4, etc., times the fundamental frequency,

A length of piano wire or other string material may be considered a rod in the above context, and therefore will resonate longitudinally at certain frequencies the values of which are basically independent ofthe diameter of the wire. The value of the resonant frequency is, however, peculiar to the material of the wire; that is a wire of the same length made of some other material than steel, having a different elastic modulus and density, may have a different fundamental longitudinal resonent frequency. Because both the density and the elastic modulus of steel piano wire normally are held constant within small tolerances, it is therefore possible to predict accurately the longitudinal resonant frequency of a length of wire once the appropriate constants are known.

When a steel piano wire is installed as a string in a piano, it has been found that the string will have a fundamental longitudinal resonant frequency whose value approaches that given by equation (2) if the speaking length of the string (the length extending between the primary terminating elements) is substituted for the value of L in the equation, even though the string itself extends beyond the string terminations to a tuning pin at one end and to an anchoring or hitch pin at the other end of the string. It has been found that the frequency of Iongitudinal vibration of such a string is essentially independent ofthe tension or pull on the string.

It has been found, however, that the actual longitudinal resonant frequency of an installed piano string is normally somewhat lower than that which would be predicted from equation (2), thereby indicating that the longitudinal effective length of the string is somewhat greater than its flexural speaking length. The relationship between the flexural speaking length ofa string and its effective length for longitudinal vibrations may be expressed L =L +D (3) wherein L is the effective length, L, is the speaking length, and D is the difference between the two lengths. For an instrument of typical construction it has been found that D is essen tially constant over a large portion of the scale. In a particular case using steel piano wire in which L and 1. were expressed in inches, the value of D was found to be approximately 1.26 inches. It must be expected, however, that the effective length of a string for longitudinal vibrations will vary with the type of string terminations in use. Therefore, the effective length of a given string must be determined by measurement for the par- DESIGNING THE SCALE 'ln'designing that portion of the instrument scale which employs plain wire strings, i.e., strings not loaded or woundwith covering wire, the frequency of the longitudinalmode of each string is tuned to the desired value by adjusting the speaking length of the string in accordance with the formula li -1. D f D (4) wherein L is the speaking length of the string, L is the effective longitudinal length, f is the desired longitudinal mode frequency, and D is the difference between the speaking length of the string and the effective length of the string for longitudinal vibrations. The value of D depends upon the characteristics of the stringterminations and the string material used and will be determined empirically. The value of E/P normally will remain constant for any specific string material.

In the section of a piano scale wherein wrapped or loaded strings are employed (as for example in the bass section of nearly all modern pianos) equation (2) cannot be used to predict the fundamental longitudinal mode frequency of a string unless it is suitably modified to take into account the effect of the loading material upon the longitudinal vibration characteristics of the string.

For wrapped strings of conventional design using a steel core wire and a copper loading wire wound helically over the core wire in a uniform continuous fashion, it has been found that the primary effect of the wrapping is to increase the apparent density of the string material without any appreciable effect upon its stiffness. It has been found that in order to calculate the fundamentallongitudinal resonant frequency of a wound string to a close approximation, it is only necessary to modify the value of P to allow for the increase in apparent density of the string material. This can be done by mutiplying P by the ratio M ,/M wherein M is the mass (or weight) per unit length of the wound string and M, is the mass (or weight) per unit length that the string would have if no loading had been'used, i.e., if the string had been made of plain wire of the same size as the core wire actually used in making the wrapped string. The values of M and M may be expressed in any system of units as long as both are expressed in the same units, and may either be computed by methods familiar to those skilled in the art or may be obtained from previously prepared wire tables usually available to the designer.

The fundamental longitudinal mode frequency ofa wrapped string may be expressed b E 1/2 b E 1/2 M 1/2 m lm .172) Marat.)-

wherein f is the fundamental longitudinal mode frequency of a wrapped string of effective length L having a core wire of elastic modulus E,. and density P where M is the mass per lineal inch of the core wire alone and Maw is the mass per lineal inch of the wrapped string. It has been found that, for practical purposes, the addition of the wrapping wire has a negligible effect on the stiffness ofthe string.

It can be seen that the quantity within the brackets inequation (5) represents the fundamental longitudinal frequency which the string would have ifthere had been no wrapping. If this frequency is previously known, the longitudinal frequency of a wrapped string of the same length can be expressed of the wrapping material necessary to tune the longitudinal mode fi'eq'ue'rrc yofa known string to the desired value.

I lt shculd also be evidentfrom eq-u ation ('5) that it is possible i to tune the frequency of the longitudinal mode of a wrapped string by varying the mass of the covering wire without changing the actual speaking length of the string. It is thus possible to achieve the correct value of longitudinal mode frequency in the bass section of an instrument even though the length of the strings is limited by the dimensions of the instrument case.

The required speaking length L for a wrapped string having a fundamental longitudinal mode frequency f is given by the equation b EB 1/2 c 1/2 1.... L... D 1 1) in which E,. and 1 comprise the elastic modulus and density of the core wire, respectively, M is the mass per unit length of the core wire only, M is the massper unit length of the wrapped string, b is a constant which depends upon the system of units employed, and D is the difference between the effective length and the speaking length of the string.

As a practical matter, D normally may be ignored. In practice, the length of the wrapped portion of a conventional wrapped string always is made less than the speaking length of the string. If this were not done there would be interference between the wrapped portion of the string and the termination elements. If the wrap were allowed to come into contact with the termination elements, damage to the wrap would occur which would tend to ruin the string. Consequently, in instruments of standard construction, the length of core wire actually wrapped is made about 1 1/4 inches less than the speaking length of the string. It would be expected, therefore, that the longitudinal mode frequency of such a wrapped string would be somewhat greater than it would be if the wrapping extended throughout the full speaking length of the string. it has been found in instruments of typical construction that the increase in longitudinal mode frequency due to the shortened wrap very nearly offsets the decrease in longitudinal mode frequency arising from the fact that the effective speaking length of the string is normally about 1 1/4 inches greater than the speaking length. Thus, to a close approximation (depending upon the particular termination) it often is possible to set the factor D equal to zero in the calculation of the speaking length of a wrapped string required to give a particular value for longitudinal mode frequency.

RELATIONSHIP BETWEEN THE TUNED FREQUENCIES It is typical of all pianos that the fundamental longitudinal frequency of vibration of a particular string is considerably greater than the frequency of the fundamental flexural mode. In nearly all conventional pianos of contemporary design using steel piano wire, the fundamental longitudinal mode frequency of a particular'string usually will have a value between 10 and 20 times the fundamental frequency of flexural vibration. This occurs for practical reasons, apart from any consideration of the longitudinal mode itself. If the string were to be stretched tightly enough so that the flexural frequency was more than about 1/10 the longitudinal frequency for a given string length, the string would be likely to break due to excessive stress on the wire. If, on the other hand, the string were to be stretched so loosely that the flexural frequency was less than about 1/20 of the longitudinal frequency, the tone quality would tend to be unacceptable by modern piano standards due to the resulting low tension and high inharmonicity.

to the present invention. lies in the fact that the mathematical relationship between the fundamental frequencies of the longitudinal and flexural modes of string vibration is held essentially constant over a considerable range of the scale. Under these conditions the relation between longitudinal and flexural fundamental vibrating frequencies can be represented as wherein f,, is the fundamental longitudinal frequency of the string, f, is the fundamental flexural frequency of the string, and n is a factor chosen by the designer to give the desired tonal result. Thus the desired value of longitudinal frequency will change if the flexural frequency of the string is changed. This dependence of the value of the desired longitudinal frequency upon the flexural frequency of the string does not imply any uncertainty or indefiniteness in the desired longitudinal frequency because the correct normal flexural frequencies of all piano strings are essentially predetermined.

As is known to those skilled in the art, the fundamental flexural frequency of any given note of the piano is, within certain tolerances, the same for all properly tuned pianos. This situation exists because in the United States and in most of the world, the frequency 440 Hz. has been established as the standard frequency to which the note A,, the 49th note, counting from the bass end of standard 88-note piano keyboard, should be tuned. The fundamental flexural frequencies of all other notes of the keyboard are derived from such a reference frequency according to the well-known relationship of the equally tempered scale in which the design value of frequency for each successive ascending note is determined by multiplying the frequency of the preceding note by the 12th root of the number 2 (which has a'value of approximately 1.05946). When the piano is tuned, the tuner, by modifying the tension exerted on the strings, adjusts the frequency of each string until the above relation between the flexural frequencies of the notes is approximately reached.

The specific relation betweenthe frequencies of fundamental, flexural and. fundamental longitudinal modes of string vibration, defined by the factor n in equation (8), is very important in the design of a piano scale according to the present invention.

In a preferred embodiment of-the invention, the factor n is chosen so that the fundamental longitudinal mode of vibration of each string will be tuned to a frequency closely corresponding to that of a note of the stretched equally tempered scale, more particularly, to the fundamental'flexural frequency of a note on the scale or keyboard of the specific instrument itself when properly tuned. For pianos, due to limitations in the tensile strength of piano wire and to restrictions imposed by current musical tastes, the useful range of n normally will be between 10 and 20. in terms of the equally tempered scale this range is equivaient to tuning the longitudinal mode within the range '4000 5200 cents sharp with reference to the fundamental flexural frequency. (in the musical context a cent is defined as l/ 100 of a semi-tone. 100 cents thus constitutes one semi-tone. An octave normally encompasses a 2:] frequency range. Because there are 12 semi-tone intervals in one octave,

1200 cents constitutes an octave. A ten to one frequency in-- terval is approximately equal to three octaves plus four semitones or 4000 cents. A frequency interval is approx'im'ately equal to four-octaves plus four semi-tones or 5200 cents.) lt it possible that either new string materials or changes in musical preference may make it desirable to extend thep'ractical range of the tuning of the longitudinal mode of string vibration outside the range 4000-5200 cents s'harp with respect to the flexural fundamental string frequency. Whether it is within the present normal range or outside this range, the preferred tuning of the longitudinal mode according to the teachings of this invention will remain at some integral multiple of 100 cents, i.e., at 3600 or at 3700 cents, or at'4000, or at 4100, or at 4200 cents, etc. I

It has been found that tuning the frequency of the longitu; dina'l mode to certain specific semi-tone or 100 cent intervals within the stated range will produce a tone quaiity more highly preferred than if it is tuned to other semi-tone intervals within a the same range. Mosthighly-preferred tunings have been found to be 4300 cents, 4400 cents, 4600 cents, 4700 cents, 4800 cents, and 5200 cents sharp. A tuning of 4000 cents is musically acceptable but results in stresses that are considered excessive for standard piano wire. In some pianos it may be desirable to avoid tuning of 4700 cents because if this tuning is used, it may tend to create a confusion of the pitch of the resultant tone because of the fact that the 4700 cents tuning produces a sound only a semi-tone removed (disregarding the octave in which the tone occurs) from the fundamental pitch of the flexural family. A tuning of 4800 cents may be preferred for some pianos, on the other hand, because the resultant tone sounds at a frequency four octaves above the fundamantal pitch of the flexural family, tending to reinforce the pitch sensation obtained therefrom. A tuning of 4600 cents has been found to produce a particularly satisfying result in certain instruments. A tuning of 4300 cents has been found to blend well with the flexural partial series and may be found to be useful in smaller instruments where excessive inharmonicity becomes a problem.

As has been previously pointed out, the exact flexural frequencies of the strings of a properly tuned piano normally differ slightly from those ofmathematically perfect equal temperament because of the inherent inharmonicity of the strings. lnharmonicity is the term given to the characteristic of a flexurally vibrating string that the frequencies of its overtones are not exact integral multiples of its fundamental frequency (as they theoretically would be if the string had no stiffness), but instead are slightly higher in frequency to a degree which depends upon the stiffness of the string and the relationship of the stiffness to other parameters. lnharmonicity influences the final flexural string frequencies of a piano during the tuning process. When the piano is tuned, the frequency of one of two notes an octave apart on the keyboard normally is adjusted with respect to the other note until a beatless condition is attained, indicating correct tuning. Those familiar with the tuning process will be aware that the condition of beatlessness" normally signifies that the second partial of the lower octave note has exactly the same frequency as the fundamental frequency of the upper note. In a string having no inharmonicity, the frequency of the second partial would be exactly twice the fundamental frequency. For actual piano strings, the frequency of the second partial normally is slightly greater than twice the fundamental frequency. Thus, the result actually achieved when two octavely-related notes are tuned for beatlessnessis that the ratio of their fundamental frequencies is slightly greater than 2:1 instead of exactly 2:1, as would be the case for mathematically perfect equal temperament.

The cumulated effect of the inharmonicity of strings throughout the scale is known as stretching" and the tuning monicity did not exist. Over the middle of the scale the average amount of stretch may be of the order of 3 cents per octave, usually increasing to a considerably larger value toward the extreme endsof the scale. Additional stretch" is often encountered in the extreme treble due to the fact thatthe tuner often estimates the correct pitch because of the difficulty of hearing beats in this part of the scale. While the amount of stretch of a properly tuned piano may vary to a certain extent with the design of the instrument, it normally will be the same for all instruments of the same design.

The effect of "stretch must be taken into account in tuning the longitudinal mode, and this may be done by tuning the lam gitudinal mode slia'rp relative to equally-tempered tuning by the same amount that a note ofthe stretched piano scale at the same frequency level is tuned sharp. For example, if it is desired to tune the longitudinal mode of the string for the lowest note on. the piano keyboard (A.,==27.5 cycles per second, nominal) to a nominal value of 4800 cents above the fundamental flexural frequency, no stretch of the longitudinal mode would be necessary because in this instance the actual I keyboard note four octaves above A is A, which is normally tuned to the reference standard pitch of 440 cycles per second, and in this octave of the piano the proper longitudinal frequency can be obtained directly from tables giving equally tempered scale frequencies. In another example, if it is desired to tune the longitudinal mode frequency of the string for the note C (flexural fundamental frequency equals 130.81 cycles per second, nominal) to a nominal tuning of 4800 cents sharp, placing the frequency of the longitudinal mode at the note C four octaves above the fundamental flexural frequency of the note, the nominal equally tempered (non-stretched) frequency of C-, would be 2093'cycles per second. However, in a typical instrument the actual C might be found to be 12 cents sharper than this value, or about 2108 cycles per second. Accordingly, the frequency of the longitudinal mode would be tuned to this latter value.

EXAMPLE OF CALCULATION OF SPEAKING LENGTH OF AN UNWRAPPED STEEL PIANO STRING elastic ino(lu1us=(20.27) dynes/ein' density =7.80 grains/em The first step is to find the desired value of longitudinal I mode frequency. Reference to a standard table ofequally tempered scale frequencies (such as A Table Relating Frequency To Cents, by Robert W. Young, Conn, 1952) indicates that the fundamental frequency of the note number 21 (F is 87.307 Hz. The same reference shows that the frequency of a note 4800 cents above this frequency is 1396.9 Hz. (the note F However, this value includes no stretch. Measurement of the flexural fundamental frequency of the note F (note number 69) on the scale ofthe instrument shows that the note is 6 cents sharp of the equally tempered value. Reference once again to Youngs tables shows that six cents sharp' corresponds for this note to a frequency of 1401.8 Hz. The frequency of the longitudinal motle should be set to this value.

Before calculating the speaking length of the string, it is necessary to determine the value of D, which will here be as- Ill The tension or pull that this string will have when it is tuned to the correct flexural frequency may be set tothe desired value by proper selection of the diameter of the string wire.

EXAMPLE OF CALCULATION FOR WRAPPED STRING To find the diameter of copper loading wire of circular cross-section required to tune the longitudinal mode frequency ola wrapped bass string to be made with 0.039 inch diameter steel piano core wire having a speaking length of 65.13 inches to a frequency of 882 Hz, assuming that the effective length of the string for longitudinal vibrations is the same as its speaking length and using E=(20.27) l0 dynes/cm and l.- =7.8 grams/cm, and also assuming that the density of the copper wrapping wire, P is 8.8 grams/cm", first rearrange equation (6) to find the mass per lineal inch of the wrapped string fi my [1 .w

25 able to solve the above expression for Muawl in terms of sumed to be 1.26 inches. The speaking length of this string may now be calculated by the use of equation (4). Ifthe units forelastic modulus and density are used as given, the constant b would have a value of unity (as it would for any consistent system of units) and the result would be in centimeters. Since it is desired to express the result in inches (1 inch =2.54 centimeters) it is necessary to assign to b the value 1/2.54

M Since it is assumed that the effective length is the same as the speaking length of the string, equation (2) may be used to find f,,,,. Because of the system of units used'to express E and P and because L, has been given in inches, the constant b must be assigned the value 1/2.54, as in the previous example.

l/2 erm t 12;)

Substituting the known frequency values in the first expression gives This means that the weight per lineal inch of the wrapped string must be 3.052 times the weight per lineal inch of the .core wire alone in order to tune the longitudinal mode frequency of this string to the desired value.

The value of M for 0.039 inch diameter steel piano wire may be computed but is more conveniently found from stanthe weight per lineal inch will be dard wire tables published by the wire manufacturer. The calculation will be given here. In the United States it has been M., =(1. 98?3) 10 Pd grains/lineal inch .For a 0.039 inch diameter steel core wire for'which P.=7*.8,

' ,=(1.9s 2 was wane- 1 =2.351 grainsllineal ine'l-i.

Substituting this value for M in the previous equation for determining M gives M =3.052(2.351) =7.17 grains/lineal inch and that the desired weight of the wrapped string is 7.17 grains per lineal inch. The weight of the desired wrapping will therefore be equal to the difference between these two values, or (7.17 2.351) 4.819 grains per lineal inch of core wire. If the diameter of the core wire is d and the diameter of the wrap is d,,, the length of wrapping wire required to make one turn around the core wire will be 11(d, +d,,.) and the numer ofturns ofwrap wire per unit length of core wire will be l/d The total length of wrap wire per unit length of core wire (assuming no space between individual turns of wrap wire) is thus The total weight of this length of wrapping wire, M is given as the product of the weight per unit length of the wrap, M times the total length of the wrap.

M I .982) l 0 P d grains/lineal inch of wrap The total weight (M,) of wrap per inch of core wire is o 114,: (1.982) 10 P d 1r 1) It should be noted that wrapping techniques affect the weight per unit-length of the finished string and that the true test of correct wrap size is whether or not the string has thecorrect longitudinal frequency when installed in the instrument.

MODIFICATIONS While the foregoing constitutes what is believed to be the best procedure for tuning the longitudinal mode of string vibration, other tuning procedures are also possible. In one such alternative procedure, the frequency of the longitudinal mode may be tuned to the frequency ofa partial ofthe flexural mode family. This procedure is less satisfactory because in the frequency range of interest (10th through 20th partials) the frequencies of string partials do not generally coincide with those of equally-tempered scale notes. Tuning to the 14th partial of the flexural mode, for example, might produce a tone Still another method of adjusting the frequency of the longitudinal mode is to tune it according to subjective pitch. This method, while productive of excellent results, presents difficulties in that subjective pitch not only varies from individual to individual but depends upon the sound level of the tone in question.

It also should be recognized that in certain rare instances it may be desirable to tune the longitudinal mode so as to deliberately produce an unpleasing, disturbing, or cacophonous sound. Specific tunings may be created to produce such results, should it be so desired.

As should now be apparent, the instant invention provides a scale so designed as to provide optimum tuning for the longitudinal mode with reference to the flexural string frequencies over a substantial range of the keyboard. It has been found that proper application of the procedures described, in conjunction with those normally employed in piano scale design, can produce instruments having improved tone quality, more even scales, and more precisely defined pitch. Changing the tuning of the longitudinal mode from one to another of the preferred intervals acts to change subtly the tone color of the instrument. Certain tone colors may be more or less preferred for certain types of music, and it is possible to produce a variety of sounds depending upon the tonal response desired by the designer.

lclaim:

l. A method of tuning a substantial number of adjoining strings of a stringed musical instrument which comprises the step of tuning both the flexural and the longitudinal modes of vibration of the said strings, the fundamental frequency of the longitudinal mode of vibration of each string so tuned being separated from the frequency of its fundamental flexural mode by an integral number of semi-tone intervals.

2. The method claimed in Claim 1 wherein both the funda mental flexural frequencies and the fundamental longitudinal frequencies of said strings are tuned to the frequencies of notes of the stretched equally tempered scale.

3. The method claimed in Claim 1 wherein the longitudinal modes of the said strings are tuned by varying the effective speaking lengths of the strings so that each such string, when excited, will vibrate longitudinally at the required frequency.

4. The method claimed in Claim 1 wherein the longitudinal modes of the said strings are tuned by varying the mass-tostiffness ratio of the strings so that each such string, when excited, will vibrate longitudinally at the required frequency.

5. The method claimed in Claim 1 wherein the particular integral number of semi-tone intervals separating the fundamental frequencies of the longitudinal and flexural modes of string vibration remains constant for the strings of a plurality of adjoining notes on the scale of the instrument.

6. A method of tuning a substantial number of consecutive strings of a piano which comprises tuning both the fle'xural and the longitudinal modes of vibration of the said strings, the fundamental flexural modes of vibration of the said strings being tuned to the frequencies of notes' of the equally tempered scale, taking into-account deviations therefrom to compensate for the added sharpness of flexural overtones resulting from the flexural inharmonicity of the strings, the fundamental longitudinal modes being tuned to frequencies within the range 4000 to 5200 cents sharp at intervals of substantially cents relative to the fundamental flexural frequencies ofthe notes to which the strings are tuned, whereby the frequency of i the fundamental longitudinal mode of a given string so tuned which would sound flat to the musical ear, while tuning to the:

16th partial of the fundamental flexural string frequency r-rray',' depending upon the individual instrument, sound out-of-tune on thes'h'arp slde because on instruments having shorter strings the frequency of the 16th partiaL'du'e to inharmonieity, is frequently much sharper than -a scale, note four octaves above the 1 e att whichiitotherwisewould, coincide.

will coincide with the frequency of the fundamental flexural mode of another string corresponding to a differentnote on the scale of the piano;

7. The method claimed in Claim 6 wherein each of' the strings of a plurality of adjacent consecutive notes has a longitudinal frequency tuned substantially the same number of cen-ts sharp with respeet to the frequency of the fundamental.

flexural mode of the string. v

8 The-method claimed in Claim 7 wherein the said strings is tuned to the frequency of a partial or overtone of the fundamental flexural mode of vibration of the string.

10. The method claimed in Claim 9 wherein the order of the chosen partial or overtone of the flexural mode to which the frequency of the longitudinal mode is tuned remains the same for a plurality of adjacent selected strings.

11. A stringed instrument in which a substantial number of consecutive strings are tuned both for flexural modes of string vibration and for longitudinal modes of string vibration, the length and mass-to-stiffness ratios of the strings being so chosen that the fundamental longitudinal mode frequencies of said strings are tuned to the frequencies of partials or overtones of the fundamental flexural string modes.

12. The instrument claimed in Claim 11 wherein the order of the chosen partial or overtone of the flexural mode to which the frequency of the longitudinal mode is tuned remains the same for a plurality of adjacent selected strings.

13. A stringed instrument in which a substantial number of consecutive strings are tuned for flexural modes of string vibration and for longitudinal modes of string vibration, the length and mass-to-stiffness ratios of the strings being so chosen that the fundamental longitudinal modes are tuned to frequencies within the range of about 40 to 52 semi-tones sharp at intervals of substantially one semi-tone relative to the fundamental flexural frequencies of the notes to which the said strings are tuned.

14. The instrument claimed in Claim 13 wherein each of the strings of a plurality of adjacent consecutive notes has a longitudinal fundamental frequency tuned substantially the same number of semi-tones sharp with respect to the frequency of the fundamental flexural mode of the string.

15. A stringed instrument in which a substantial number of consecutive adjoining strings are tuned both for flexural modes of string vibration and for longitudinal modes of string vibration, the strings being tensioned to tune their fundamental flexural modes essentially to the frequencies of notes of the equally tempered scale, the lengths and mass-to-stiffness ratios of the strings being such that their fundamental longitudinal modes are also tuned essentially to the frequencies of the equally tempered scale, the fundamental frequencies of both flexural and longitudinal modes deviating from exact equal temperament by an amount just sufficient to compensate for the frequency shift of flexural overtones caused by the flexural inharmonicity of the strings. A

16. The stringed instrument claimed in Claim 15 wherein each of said strings has a tuned fundamental longitudinal frequency substantially no greater than 5000 Hz.

17. A piano in which the adjoining strings in at least about the lower half of the scale of the instrument are tuned both for flexural modes of string vibration and for longitudinal modes of string vibration, said adjoining strings being tensioned to tune their fundamental flexural modes of string vibration to the frequencies of notes of the equally tempered scale, including such frequency deviations as are necessary to take into account the effects of the flexural inharmonicity of the strings, each of said adjoining strings having a length and mass-to-stiffness ratio such that the frequency of its fundamental longitudinal mode of string vibration is tuned approximately (u) cents sharp relative to its fundamental flexural frequency, (u) bemg any'integer between 40 and 52, the exact value ot the frequency of the longitudinal mode of each such tuned string being set to coincide substantially with the frequency of the fundamental flexural mode of vibration of other strings within the piano whose fundamental flexural modes are tuned (u) semi-tone intervals sharp with respect to the fundamental flexural frequency of each such tuned string.

18. The piano claimed in claim 17 wherein the value of (:1) remains the same for a plurality of adjoining strings. 

